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Chapter 6: 34 E (page 265)

a. For an invertiblen脳n matrixA and an arbitraryn脳n matrix B, show that rref[A鈭�AB]=[In鈭�B]rref[A鈭�AB]=[In鈭�B].Hint: The left part ofrref[A鈭�AB] is rref(A)=In. Write rref [A鈭�AB]=[In鈭�M]; we have to show thatM=B . To demonstrate this, note that the columns of matrix [B-In]are in the kernel of[A鈭�AB] and therefore in the kernel of [In鈭�M].
b. What does the formula rref[A鈭�AB]=[In鈭�B]tell you if B=A-1?

Short Answer

Expert verified

Therefore,

a) Hence it鈥檚 provedM=B 15

b) The formula said to find the inverse of an invertible.

Step by step solution

01

To prove M=B

SinceA is invertible, thenrrefA=In.

So there exists ann脳nmatrixM such that

rref[A鈭�AB]=In鈭�M

We have to prove thatM=B.


We notice that the columns of the matrix.

B鈭�In

Are in the kernel ofrref[A鈭�AB], therefore also in the kernel ofrrefIn鈭�M. So we compute:

InMB鈭�In=B鈭�M

InMB鈭�In=0 , which gives us M=B.

02

What does the formula rref[A∣AB]=[In∣B]  tell you if B=A-1

IfB=A鈭�1, we haverrefA鈭�In=In鈭�A鈭�1.

This means we can use this very method to find the inverse of an invertiblen脳n matrix A.

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